Adding new Representations of Mathematical Objects to Aplusix

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Adding new Representations of Mathematical Objects to Aplusix

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Adding new Representations of Mathematical Objects to Aplusix

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Adding new Representations of Mathematical Objects to Aplusix

Denis Bouhineau, Hamid Chaachoua, Jean-Francois Nicaud & Christophe Viudez

1

ICTMT’2007

What’s next ?

- The ReMath project
- Natural representation of algebraic expression in Aplusix
- Tree & Natural representation of algebraic expression in Aplusix
- Motivations
- Questions raised
- Answers
- Experiments

- Graphical representations of algebraic expression in Aplusix
- Conclusion

2

ICTMT’2007

The ReMath project

- The ReMath project (IST4-26751 European project, Dec 2005)
- Representing Mathematics with Digital Media
- ITD-CNR (Genova), NKUA – ETL, Talent S.A (Athens), UNISI (Sienna), METAH (Grenoble), Didirem (Paris), LKL-UOL (London)

- Objectives
- Enrich state-of-the-art dynamic digital artefacts for doing mathematics with new representations of mathematical objects
- Work on scenarios for the use of these artefacts
- Carry out empirical research involving cross-experiments in realistic educational contexts

3

ICTMT’2007

Natural representation of algebraic expressions in Aplusix

- Aplusix
- A microworld and an exerciser for doing algebra
- Students freely write algebraic expressions

- Algebraic expression
- Natural representation of algebraic expressions
- Natural editing of algebraic expressions

4

ICTMT’2007

Natural representation of algebraic expressions in Aplusix

- Representation of the reasoning processes with a tree
- Two fundamental feedbacks
- Semantic equivalence between successive steps
- Syntax of the final expression

- Users (students)
- Gain autonomy
- Learn algebra

- Available
- for research, http://aplusix.imag.fr/Dir-Vers-Rech
- or see publishers : Chartwell&Yorke (uk), Les éditions Archimède (fr), MediaDirect (it)

5

ICTMT’2007

Tree & Natural representation of algebraic expression in Aplusix

- Motivation (ideal)
- epistemological : trees are natural representations of algebraic expressions
- didactical :
- introduction of trees = change of register
- mapping between natural & tree object understand the syntactical structure of algebraic expression

- computer science : trees are fundamental objects

- Motivation (pragmatic)
- ReMath
- Didactician colleague’s ask

6

ICTMT’2007

Tree & Natural representation of algebraic expression in Aplusix

- Questions about the kind of tree :
- internal trees used by Aplusix ?
- special algebraic trees ?
- abstract trees ?

- Questions about the link between tree representation and natural representation :
- just a way to display object / edit ?
- ill-formed ?

- Mathematical questions :
- ‘-’ operator ?
- ‘(‘ and ‘)’ ?

7

ICTMT’2007

Tree & Natural representation of algebraic expression in Aplusix

- Answers
- authentic objects of our microworld
- abstract trees
- 4 modes (representation)
- natural
- mixed
- free mode
- controlled mode

- Other answers (mode)

- First prototype : Dec 2006

8

ICTMT’2007

Tree & Natural representation of algebraic expression in Aplusix

- New sorts of exercise
- build the tree representation of an expression given in the natural representation
- build the natural representation of an expression given in the tree representation

- Experiments
- planed in France and italie in late 2007

9

ICTMT’2007

Graphical representations of algebraic expression in Aplusix

- Objective : only display
- Motivations
- asked by teachers
- present in curriculum
- combining symbolic and graphical representations

- Questions raised
- How to represent the solution of equations ?
- How to represent identical objects ?

10

ICTMT’2007

Conclusion

- Adding new Representations of Mathematical Objects
- Decide whether the representation will be an object or just a new way of displaying object
- Think about experiments and use cases (à la UML)
- Work with colleagues from other laboratories and different cultures
- (plan time enough for debugging !)

11

ICTMT’2007